Question

1. 9k² + 9k - 4 (and a boxed multiplication-like structure with terms 3k, 4, k, -3k, -4, and a factored form (3k - 1)(3k + 4) maybe)

Explanation:

Step1: Analyze the quadratic expression

We have the quadratic expression \(9k^{2}+9k - 4\). We can try to factor it using the method of splitting the middle term or by using the box method (as shown in the image).

Step2: Find two numbers that multiply to \(9\times(- 4)=-36\) and add up to \(9\)

The two numbers are \(12\) and \(- 3\) since \(12\times(-3)=-36\) and \(12+( - 3)=9\).

Step3: Rewrite the middle term

We rewrite the middle term \(9k\) as \(12k-3k\). So the expression becomes:
\(9k^{2}+12k - 3k-4\)

Step4: Group the terms

Group the first two terms and the last two terms:
\((9k^{2}+12k)-(3k + 4)\)

Step5: Factor out the common factors from each group

From the first group \(9k^{2}+12k\), we can factor out \(3k\): \(3k(3k + 4)\)
From the second group \(-(3k + 4)\), we can factor out \(- 1\): \(-1(3k + 4)\)

Step6: Factor out the common binomial factor

Now we have \(3k(3k + 4)-1(3k + 4)\). We can factor out \((3k + 4)\) from both terms:
\((3k - 1)(3k+4)\)

Answer:

The factored form of \(9k^{2}+9k - 4\) is \(\boldsymbol{(3k - 1)(3k + 4)}\)